Theory question 110
Measures of DispersionThe table below shows the distribution of the waiting times for some customers in a certain petrol station.
Waiting time (in min.): 1.5 − 1.9, 2.0 − 2.4, 2.5 − 2.9, 3.0 − 3.4, 3.5 − 3.9, 4.0 − 4.4
No. of Customers: 3, 10, 18, 10, 7, 2
(a) Write down the class boundaries of the distribution.
(b) Construct a cumulative frequency curve for the data.
(c) Using your graph, estimate: (i) the interquartile range of the distribution; (ii) the proportion of customers who would have waited for more than 3 minutes.
Show the model answer
(a) 1.45−1.95, 1.95−2.45, 2.45−2.95, 2.95−3.45, 3.45−3.95, 3.95−4.45
(b) Cumulative frequencies 3, 13, 31, 41, 48, 50 plotted against the upper boundaries 1.95, 2.45, 2.95, 3.45, 3.95, 4.45 (start at (1.45, 0)); join with a smooth S-shaped curve
(c)(i) Q1 (12.5th) ≈ 2.4 min, Q3 (37.5th) ≈ 3.3 min; interquartile range ≈ 0.8−0.9 min
(ii) About 32 customers waited up to 3 minutes, so about 18 of 50 waited longer: 18/50 = 0.36 (36%)
Plot cumulative frequency at the upper class boundary, not at the class mark.